The Generalized Entropy Ergodic Theorem for Nonhomogeneous Markov Chains

Let $({\xi }_n{)}_{n=0}^{\mathrm{\infty }}$ be a nonhomogeneous Markov chain taking values from finite state-space of $\mathbf{X}=\{1,2,\dots ,b\}$. In this paper, we will study the generalized entropy ergodic theorem with almost-everywhere and ${\mathcal{L}}_1$ convergence for nonhomogeneous Markov chains, which generalizes the corresponding classical results for the Markov chains.